Evgeny Sklyanin

On Bäcklund transformations for many-body systems

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Backlund transformations (BTs) from the Hamiltonian point of view. The analogy between BT and Baxters quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variablefor the parameter λ of the BT Bsuch thatbelongs to the spectrum of the Lax operator L(λ). As a consequence, the generating function of the composition B�1 ◦ . . . ◦ Bn of n BTs gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized byis introduced.

Quantum Bäcklund transformation for the integrable DST model

For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analog of the Backlund transformation (Q-operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q-operator as an explicit integral operator as well as describe its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The found integral equations are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analogue of the Backlund transformation (Q -operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q -operator as an explicit integral operator as well as describing its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The integral equations found are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

Generating Function of Correlators in the sl2 Gaudin Model

An exponential generating function of correlators is calculated explicitly for the sl2 Gaudin model (degenerated quantum integrable XXX spin chain). The calculation relies on the Gauss decomposition for the SL(2) loop group. A new explicit expression for the correlators is derived from the generating function, from which the determinant formulas for the norms of Bethe eigenfunctions due to Richardson and Gaudin are obtained.

Q-operator and factorised separation chain for Jack polynomials

Abstract Applying Baxters method of the Q -operator to the set of Sekiguchis commuting partial differential operators we show that Jack polynomials P λ (1/g) ( χ 1 , …, χ n ) …, χ n ) are eigenfunctions of a one-parameter family of integral operators Q z . The operators Q z are expressed in terms of the Dirichlet-Liouville n -dimensional beta integral. From a composition of n operators Q zk we construct an integral operator S n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S n admits a factorisation described in terms of restricted Jack polynomials P λ (1/g) ( x 1 , …, x k , 1, … 1). Using the operator Q z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.

Integrable Hierarchy of the Quantum Benjamin-Ono Equation

A hierarchy of pairwise commuting Hamiltonians for the quantum periodic Benjamin{Ono equation is constructed by using the Lax matrix. The eigenvectors of these Hamiltonians are Jack symmetric functions of infinitely many variables x1;x2;:::. This construction provides explicit expressions for the Hamiltonians in terms of the power sum symmetric functions pn = x n + x n + and is based on our recent results from (Comm. Math. Phys. 324 (2013), 831{849).

Bispectrality for the quantum open Toda chain

An alternative to Babelon (2003 Lett. Math. Phys. 65 229) construction of dual variables for the quantum open Toda chain is proposed that is based on the 2 × 2 Lax matrix and the corresponding quadratic R-matrix algebra.

Sekiguchi-Debiard Operators at Infinity

We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables

Macdonald operators at infinity

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